Exploiting the Link Between the Wilcoxon-Mann-Whitney Test and a Simple Odds Parameter

• Activity Number: 241
• Type: Invited
• Date/Time: Tuesday, August 9, 2005 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Consulting
• Author(s): Ralph O'Brien*+ and John Castelloe
• Companies: Cleveland Clinic Foundation and SAS Institute, Inc.
• Keywords: Wilcoxon rank-sum test ; Mann-Whitney U test ; effect size ; generalized odds ratio ; power ; sample-size analysis

Abstract

The Wilcoxon-Mann-Whitney (WMW) test often is said to compare two independent medians, but this is only valid under conditions rarely met in practice. So, what does it test? Let Y1 and Y2 be sample values from two independent groups, and pi = Prob[Y1 > Y2] + Prob[Y1 = Y2]/2. This is the nonparametric area under the ROC curve in diagnostic testing, a field that routinely estimates and forms CIs for pi and tests H0: pi = 0.50. With no ties, theta = pi/(1 - pi) is a special case of the generalized odds ratio by Agresti (1980), who gave asymptotic standard errors for estimates of theta and log(theta), thus providing CIs and a test for H0: theta = 1.0. Generalizing theta to allow ties gives us a meaningful parameter and CI to augment the ubiquitous WMW p value. This also gives us a new way to handle sample-size analyses, competing with Kolassa's (1995) solution. We assess these methods using Monte Carlo studies and illustrate them with a two-arm clinical trial involving a seven-point Likert measure. Finally, we show perplexities that can result when the WMW test is applied to ordered categorical data, thus confirming it does not necessarily compare central tendencies.